In terms only of forces acting ON an
object or system, we have the following:
1. The object or system can be subjected to any number of forces acting ON the
system. The net force F_net_ON is the sum, the net effect, of all those forces.
2. On any given interval the work done by the net force is equal to the change
in the KE of the object or system.
3. This is summarized in the work-kinetic energy theorem
`dW_net_ON = `dKE
4. Each force acting on the object or system can be classified as some
combination of conservative and nonconservative forces, so
5. the net force can be expressed as the sum of a net conservative and a net
nonconservative force:
F_net_ON = F_net_cons_ON + F_net_noncons_ON.
6. Thus `dW_net_ON = `dW_net_cons_ON + `dW_net_noncons_ON.
7. Change in PE can be defined to be equal and opposite the work done ON the
system by conservative forces:
`dW_net_cons_ON = - `dPE
8. Since `dW_net_ON = `dW_net_cons_ON + `dW_net_noncons_ON, the work-kinetic
energy theorem becomes
`dW_net_cons_ON + `dW_net_noncons_ON = `dKE.
9. Since `dW_net_noncons_ON = -`dPE this can be written
-`dPE + `dW_net_noncons_ON = `dKE.
10. This can be rearranged to
`dW_net_noncons_ON = `dKE + `dPE.
In the above we have explained the relationships among six quantities:
`dKE
F_net_ON
`dW_net_ON
`dW_net_ON_cons
`dW_net_ON_noncons
`dPE
The main relationships are
`dW_net_ON = `dKE and
`dW_net_ON_noncons = `dKE + `dPE.
If we replace the word ON by the word BY (indicating forces exerted and work done BY rather than ON the system), the force and therefore the work reverse sign. In particular this gives us
`dKE + `dPE + `dW_net_BY_noncons = 0,
a form which is useful in understanding
some problems.
You may apply these principles to the suggested exercises below:
Suggested exercise:
If you choose to do all or part of this exercise, copy these questions starting
with this line and insert any responses in the lines between the @#@# and #$&* marks. You
may submit it using the Submit Work Form, with the title 'Identifying
Work-Energy Quantities'.
In each of the following situations, identify the events that begin and end the
obvious interval. For this interval, identify each of the six quantities, then
apply one or both of the two main relationships to the situation to explain how
the six quantities are related. One or more of the quantities may in some
situations be either zero, or small enough to be considered negligible. Be sure
to specify the direction that will be considered positive, then state whether
each force you consider is in the positive or negative direction and whether the
work done by this force is positive or negative.
Situation A: I very slowly raise a 35-pound concrete block from the ground to a
point over my head, two meters above the ground. After identifying the interval,
identify the six quantities from the point of view of my hands (i.e., identify
the forces which act on my hands). Use the two main relationships to explain how
these quantities are related.
@#@#
#$&*
Situation B: I very quickly raise a 35-pound concrete block from the ground to a
point over my head, two meters above the ground, at which point it is moving
fairly quickly in the upward direction. After identifying the interval, identify
the six quantities from the point of view of the block (i.e., consider the
forces acting on the block). Use the two main relationships to explain how these
quantities are related.
@#@#
#$&*
Situation C: I hook one end of a rubber band to a rigid support, and pull back
on the other with me hand, stretching the rubber band from the length where it
begins to experience tension to 30% greater than this length. Consider this from
my perspective (i.e., consider the forces exerted on my fingers at the point
where they grasp the rubber band).
@#@#
#$&*
Situation D: I hook one end of a rubber band to a rigid support, and pull back
on the other with me hand, stretching the rubber band from the length where it
begins to experience tension to 30% greater than this length. Consider this from
the perspective of the rubber band, at the end where I exert the force (i.e.,
consider the forces exerted on that end of the rubber band).
@#@#
#$&*
Situation E: I hook one end of a rubber band to a block on a tabletop, and pull
back on the other with me hand, in a direction parallel to the tabletop. The
rubber band stretches by 20% before the block begins to slide, then the block
slides from one end of the table to the other. Starting when the rubber band is
at a length where it begins to experience tension and ending when it reaches the
edge of the tabletop. Consider this from my perspective.
@#@#
#$&*
Situation F: I throw a ball upward. Consider the interval between the ball's
release and the ball's contact with the ground, from the perspective of the
ball.
@#@#
#$&*
For each situation, you have identified the forces acting ON some point, from
the perspective of one object or the other.
Now see if you can identify the forces exerted BY the various objects:
Situation A': Identify the forces exerted BY my hands, rather than ON my hands,
and the work done by those forces.
@#@#
#$&*
Situation B': Identify the forces exerted BY the block, rather than ON the
block, and the work done by those forces.
@#@#
#$&*
Situation C': Identify the forces exerted By my fingers, rather than ON my
fingers, and the work done by those forces.
@#@#
#$&*
Situation D': Identify the forces exerted By the end of the rubber band, rather
than ON the end, and the work done by those forces.
@#@#
#$&*
Situation E': Identify the forces exerted By my fingers, rather than ON my
fingers, and the work done by those forces.
@#@#
#$&*
Situation F': Identify the forces exerted By the end of the rubber band, rather
than ON that end, and the work done by those forces.
@#@#
#$&*